Quantitative prediction of 3D solution shape and flexibility of nucleic ...

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Quantitative prediction of 3D solution shape and flexibility of nucleic ...

2862–2868 Nucleic Acids Research, 2012, Vol. 40, No. 7 Published online 10 December 2011

doi:10.1093/nar/gkr1173

Quantitative prediction of 3D solution shape

and flexibility of nucleic acid nanostructures

Do-Nyun Kim 1 , Fabian Kilchherr 2 , Hendrik Dietz 2, * and Mark Bathe 1, *

1 Laboratory for Computational Biology & Biophysics, Department of Biological Engineering, Massachusetts

Institute of Technology, Cambridge, MA 02139, USA and 2 Department of Physics & Walter Schottky Institute,

Center for Integrated Protein Science, Technische Universität München, Garching, Germany

Received September 13, 2011; Revised November 10, 2011; Accepted November 11, 2011

ABSTRACT

DNA nanotechnology enables the programmed synthesis

of intricate nanometer-scale structures for

diverse applications in materials and biological

science. Precise control over the 3D solution

shape and mechanical flexibility of target designs

is important to achieve desired functionality.

Because experimental validation of designed

nanostructures is time-consuming and costintensive,

predictive physical models of nanostructure

shape and flexibility have the capacity to

enhance dramatically the design process. Here, we

significantly extend and experimentally validate

a computational modeling framework for

DNA origami previously presented as CanDo

[Castro,C.E., Kilchherr,F., Kim,D.-N., Shiao,E.L.,

Wauer,T., Wortmann,P., Bathe,M., Dietz,H. (2011) A

primer to scaffolded DNA origami. Nat. Meth., 8,

221–229.]. 3D solution shape and flexibility are predicted

from basepair connectivity maps now accounting

for nicks in the DNA double helix,

entropic elasticity of single-stranded DNA, and

distant crossovers required to model wireframe

structures, in addition to previous modeling

(Castro,C.E., et al.) that accounted only for the

canonical twist, bend and stretch stiffness of

double-helical DNA domains. Systematic

experimental validation of nanostructure flexibility

mediated by internal crossover density probed

using a 32-helix DNA bundle demonstrates for the

first time that our model not only predicts the

3D solution shape of complex DNA nanostructures

but also their mechanical flexibility. Thus, our

model represents an important advance in the

quantitative understanding of DNA-based

nanostructure shape and flexibility, and we

anticipate that this model will increase significantly

the number and variety of synthetic nanostructures

designed using nucleic acids.

INTRODUCTION

Programmable self-assembly of complementary singlestranded

nucleic acids is a versatile approach to designing

sophisticated nanoscale structures (2–4). Scaffolded DNA

origami is a successful approach that enables construction

of DNA-based nanostructures in which hundreds of

‘staple’ oligonucleotides hybridize with a ‘scaffold’

strand that is several-thousand-bases long. This structure

then consists of numerous double-helical domains linked

via covalent phosphate linkages into a complex 3D

geometry (1,5–11).

Designing DNA nanostructures to meet structural and

mechanical specifications is a formidable task even for

experts in the field due largely to the lack of predictive

computational design frameworks. For example, slight

geometrical incompatibilities between parallel doublehelical

DNA domains superpose to induce global

shape deformations including twist and bend (7). While

such deformations are intuitive for simple shapes, they

become highly non-intuitive when heterogeneous combinations

of twist-stretch-bend are present. Further,

nanostructures are subject to significant thermal forces

in solution that disorder the structure. Accounting quantitatively

for thermally induced shape fluctuations is important

to applications requiring nanometer-scale control

over target shape. Given the substantial time and financial

cost associated with the manual design and experimental

validation of designed nanostructures, predictive computational

tools are of great use for increasing the number

and variety of DNA-based nanostructures produced.

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*To whom correspondence should be addressed. Tel: +1 617 324 5685; Fax: +1 617 324 7554; Email: mark.bathe@mit.edu

Correspondence may also be addressed to Hendrik Dietz. Tel: +49 89 289 11615; Fax: +49 89 289 11612; Email: dietz@tum.de

ß The Author(s) 2011. Published by Oxford University Press.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/

by-nc/3.0), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.


Nucleic Acids Research, 2012, Vol. 40, No. 7 2863

To meet this need, we previously developed a computational

modeling framework to predict the 3D solution

shape and flexibility of DNA-based nanostructures (1).

While this model accounted for the canonical bend,

twist, and stretch stiffness of the DNA double helix, it

ignored important effects including backbone nicks in

DNA strands, entropic elasticity of single-stranded DNA

used to design, for example, tensegrity structures (11),

and distant crossovers used in wireframe structures (7).

Here, we significantly extend this modeling framework

to include these features of DNA nanostructure designs,

and systematically validate experimentally its prediction

of nanostructure solution shape and flexibility.

METHODS

Negative-stain TEM image analysis

Samples were applied to glow-discharged (Glow discharger

EMS 100, Electron Microscopy Sciences)

carbon-coated TEM grids (EFCF400-Cu-50, Electron

Microscopy Sciences) and negative-stained with a 2%

uranyl formate stain solution including 0.5% 5 M

sodium hydroxide solution. Transmission electron micrographs

were taken with a Philips CM100 transmission

electron microscope operated at 100 kV. Single particle

libraries were generated from micrographs taken at a

magnification of 28 500 by boxing individual particles

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Figure 1. Prediction of 3D solution shape of non-linear DNA-based nanostructures. (a) DNA double helices are treated as isotropic elastic rods

connected by rigid crossovers (left). Spheres correspond to finite element nodes used to describe the position and orientation of DNA basepairs,

and lines represent finite element beams that model the stretching, twisting and bending mechanics of DNA. 3D solution shape is computed using a

mechanical energy perturbation in which double helices are first deformed via stretching and twisting until crossover locations are in register,

followed by energy minimization (right). The undeformed, deformed and mechanically relaxed (equilibrium) conformations of the DNA

nanostructure are denoted U, D and R in the schematic at right. (b) Example deformations induced by mismatch between neighboring DNA

helices induced by (red) insertions and (blue) deletions for a honeycomb lattice bundle (Supplementary Figures S2 and S17). (c) Snapshots of a

spherical wireframe structure (7) during the deformation and relaxation process.


2864 Nucleic Acids Research, 2012, Vol. 40, No. 7

with the boxer routine of EMAN2 (12). Particles with

obvious defects were omitted. Second, a reference

particle was chosen out of the library and each individual

particle of the library was aligned against the reference

particle using the multi-ref-align routine of IMAGIC

4.0 (Imagic software, Image Science Software GmbH,

Berlin, Germany). The new libraries with aligned particles

were re-inspected and incorrectly aligned particles were

removed. Finally, average and variance images were

obtained with the sum-images routine of IMAGIC 4.0.

Computational model for nucleic acid shape

and mechanics

DNA origami nanostructures are modeled as bundles of

isotropic elastic rods that are rigidly constrained to their

nearest neighbors at specific crossover positions. Each rod

consists of a set of two-node beam finite elements representing

the stretching (1 100 pN), bending (230 pN nm 2 ),

and torsional (460 pN nm 2 ) stiffness of the B-form DNA

double helix. Nicks are modeled by reducing backbone

bending and torsional stiffness by a factor of 100

whereas stretching stiffness is retained at double-helix

values. The value of 100 was chosen empirically based

on a comparison of solution shapes with extensive nicks

with experimental TEM images, and is also consistent

with the base-stacking free energy of DNA (13). A more

sophisticated model might employ distinct stiffnesses for

the unruptured and ruptured states of the double-helix

backbone instead of a single, reduced backbone stiffness,

which may be the subject of future work. The position and

orientation of DNA basepairs are represented by beam

finite element nodal degrees of freedom. Connections

between double helices formed from single-stranded

DNA are treated as non-linear springs that model the

force-extension behavior of the modified freely jointed

chain model (14) (Supplementary Note 1). This approximation

is valid for worm-like chains with a persistence

length that is considerably smaller than the chain

contour length. Distant crossovers used in wireframe

structures are modeled as stiff beams that gradually

shrink during the perturbation analysis until their length

reduces to the distance between neighboring helices or the

helix diameter. Relative orientations of helices connected

by distant crossovers are unconstrained so that our model

is limited at present to on-lattice modeling of DNA

origami structures. The mechanical perturbation analysis

used to compute the equilibrium solution shape begins

with the undeformed conformation of the DNA origami

structure designed on a honeycomb or square lattice.

Helices are arranged in parallel on the lattice using

standard B-form DNA geometric parameters (helix

diameter of 2.25 nm, helicity of 10.5 basepairs per turn,

and axial rise of 0.34 nm per basepair). The structure is

initially deformed as shown in Figure 1 to eliminate axial

and torsional mismatch between neighboring helices, and

subsequently rigid constraints are applied to fully constrain

all degrees of freedom of neighboring helices at

crossover locations. The final solution shape

is computed iteratively using the commercial finite

element software program ADINA (Automatic Dynamic

Incremental Non-linear Analysis, ADINA R&D Inc.,

Watertown, MA, USA) using a geometrically non-linear

finite element formulation to reach mechanical equilibrium.

Normal mode analysis (15,16) is performed at this

mechanical free energy minimum to compute the mechanical

flexibility of the folded structure. Root-mean-square

fluctuations of basepairs are computed at 298 K for the

final solution shape using 200 lowest normal modes and

the equipartition theorem of statistical thermodynamics

(17). The time required to compute these deformed

shapes ranges from 5 to 35 min on an Intel Õ Core TM i7

processor with 12 GB RAM.

Double-helical DNA is treated as a homogeneous, isotropic

elastic rod. Sequence-dependent mechanical

properties (18), interhelical electrostatic repulsion, and

the major–minor groove of DNA are ignored. Singlestranded

DNA connecting double-helical domains is

treated as an entropic spring (14) (Supplementary

Note 1) and nicks in DNA strand backbones are

modeled using reduced bending and torsional stiffness

compared with non-nicked double-helical DNA

(Figure S1). Phosphate backbone strand crossovers are

modeled as rigid links of zero length (Figure 1a). To

compute the 3D shape of the DNA nanostructure, the

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Figure 2. Effect of nicks on 3D solution shape. (a) Snapshots of ‘A’-like object during the deformation and relaxation process. (b and c) Predicted

solution shapes of an ‘A’-like object (b) including the nick model and (c) ignoring the nick model.


Nucleic Acids Research, 2012, Vol. 40, No. 7 2865

Figure 3. Experimental validation of non-linear solution shapes of DNA nanostructures. (a and b) TEM images (left) and predicted solution

shapes (middle and right) for objects resembling capital letters ‘A’ and ‘S’ (Supplementary Figures S18–S19). TEM images are obtained by averaging

single particle images (Supplementary Figures S4–S5). (c) Predicted solution shapes of a tensegrity structure (11). (d) Predicted solution shapes of a

single-layer rectangular structure (5).

procedure performs a mechanical perturbation analysis in

which DNA domains are simultaneously relaxed from

their initial structure, assumed to reside on a honeycomb

or square lattice, to their minimum mechanical free energy

shape (Figure 1a). To achieve this, neighboring helices are

first deformed in the axial and torsional directions to align

strand crossovers assuming an average B-form DNA

helicity of 10.5 bp per turn (19). Crossover constraints

are then applied between neighboring helices and the

structure is relaxed iteratively using a finite element

approach that is well established in the structural mechanics

field (20). The initial structure assumes internal

registry of backbone positions compatible with the

honeycomb-type or square-lattice packing (6,7).

RESULTS AND DISCUSSION

Application of the procedure to a simple 6-helix

honeycomb unit cell of 178-basepair length illustrates

its ability to predict intuitive shapes induced by

site-directed mismatches in crossover positions (Figure

1b and Supplementary Figure S2). Application to a

complex 3D wireframe structure that involves highly

non-linear mechanical energy perturbations illustrates

the model’s predictive power when non-intuitive shape

distortions are accompanied by significant local buckling

and distortions along the computed mechanical perturbation

pathway (Figure 1c and Supplementary Figure S3).

Such calculations of the solution shapes of wire-frame

structures including distant crossovers were impossible

with our previous model (1).

To further explore its ability to predict wireframe

topology shapes, we built a novel multi-layer DNA

origami structure that resembles a capital letter ‘A’

(Figures 2, 3a and Supplementary Figure S4). This

object has a complex wireframe topology with a final

structure that is distant from the initial structure used

for the perturbation analysis in our model. The model

correctly predicts the global shape within obvious experimental

error, despite the high non-linearity involved in the

mechanical perturbation pathway (Figure 2a). This shape

illustrates the importance of modeling nicks because the

symmetric ‘A’ shape (Figures 2b and 3a) observed experimentally

is not obtained when nicks are ignored

(Figure 2c). We also designed and built a novel multi-layer

DNA origami structure resembling the capital letter

‘S’ (Figure 3b and Supplementary Figure S5), which

includes two domains of opposite curvature. Again, the

model correctly predicts the global structure including

a precise match of curvature within experimental

error. Importantly, the model also predicts an out-ofbending-plane

twist deformation that is unfortunately invisible

in the experimental data because ‘S’ particles

adhere to the TEM support surface in the flat orientation

shown in Figure 3b. The object’s curvature was designed

by balanced insertions and deletions that cancel

right-handed and left-handed torques induced by local

over- and underwinding in the structure (7). However,

this computational result suggests that these torques are

in fact unbalanced because of distinct internal torques

of double-helical DNA segments of different length

that are over- or underwound by the same twist angle.

This result illustrates a non-intuitive feature of 3D shape

that is likely to be present in solution but may remain

unnoticed experimentally due to limitations in imaging

3D shape.

DNA origami objects packed on the square lattice have

been shown to exhibit global twist deformation in the

absence of insertions and deletions due to underwinding

of double helices with an average helicity of 10.67 bp

per turn (21). This is in contrast to honeycomb lattice

structures that appear undeformed when crossovers

are spaced at intervals of 10.5 bp per turn (7). To test

the ability of our model to predict the influence of

lattice geometry on overall shape, we analyzed a set of

multi-layer DNA origami blocks designed on a square

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2866 Nucleic Acids Research, 2012, Vol. 40, No. 7

Figure 4. Experimental validation of the flexibility of DNA nanostructures. (a) 32-helix bundle objects packed on a honeycomb lattice

with decreasing crossover density corresponding to axial spacings of 21 (1st column), 42 (2nd column), 63 (3rd column) and 84 (4th column)

basepairs (Supplementary Figures S20–23). The inset depicts the three distinct crossover orientations in the honeycomb lattice. (b) Experimental (top)

and simulated (middle) TEM images obtained from single-particle averaging (Supplementary Figures S10–S14), and (bottom) quantitative distributions

in root-mean-square fluctuation amplitude. (c) 3D renderings and quantitative distributions in root-mean-square fluctuation amplitude of

robot-like object (2nd column), letter ‘A’ (3rd column), and letter ‘S’ (4th column). TEM image of robot-like object (1st column) is obtained

by averaging single particle images (Supplementary Figure S15).

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lattice (21). In agreement with these published data, our

model predicts a global twist deformation, with decreasing

twist angle that results from the increasing torsional stiffness

of blocks of increasing thickness (Supplementary

Figure S6). Interestingly, for a single-layer rectangular

design, the predicted shape exhibits overall bend in

addition to twist due to the non-linear stretch-bend-twist

energetic coupling that emerges at large twist angle

(Figure 3d and Supplementary Figures S7–S8).

Importantly, this coupling is not due to intrinsic mechanical

coupling of these modes of deformation of the DNA

double-helix itself, but rather the mechanical coupling of

these degrees of freedom resulting from rigid crossovers

that connect neighboring helices in the structure. 3D

cryo-electron microscopy of free-standing structures will

be required to validate this subtle model prediction. We

validated the ability of our model to predict solution

shapes more generally for a variety of complex shapes


Nucleic Acids Research, 2012, Vol. 40, No. 7 2867

including a previously published tensegrity motif (11),

which, like the preceding wireframe structures, could not

be analyzed using our previous modeling framework (1).

Importantly, shape predictions are made correctly in a

completely automatic, unsupervised manner without any

feedback from the designer (Supplementary Figures S9

and S16), greatly enhancing the design–test–design cycle

due to the elimination of the financially costly and

time-consuming experimental validation step.

Thermal forces in solution tend to disorder nucleic

acid nanostructures, potentially eliminating their ability

to fulfill target function. The degree to which structural

features match their target shape in solution, termed here

‘mechanical integrity’, is therefore a centrally important

property to consider in nanostructure design. Notably,

this feature of DNA origami structures is distinct from

the question of folding stability of the structure, because

it is concerned specifically with the question of what the

flexibility or compliance of the folded, target structure is.

One tunable structural parameter that is expected to influence

mechanical integrity directly is the density of phosphate

backbone crossovers that link double-helical DNA

domains. To test the ability of our model to predict the

effect of this design parameter on flexibility or integrity,

we designed and built four versions of a 32-helix

multi-layer DNA origami object in honeycomb-lattice

packing with decreasing densities of crossovers linking

neighboring double-helical domains (1 crossover per

21, 42, 63 and 84 bp) (Figure 4a). Negative-stain TEM

micrographs illustrate the conformational variability of

the resulting objects (Supplementary Figures S10–S14).

Superposition of single-particle images illustrates the

effect of decreasing crossover spacing on mean object

conformation (Figure 4b), with objects becoming more

polymorphic structurally as crossover density decreases.

We extended our framework to compute ‘mean fluctuation

micrographs’ due to thermal fluctuations at 298 K to

enable direct comparison with experiment. Again, visual

correspondence within experimental error is achieved

(Figure 4b). In contrast to experiment, however, the

model enables direct quantitation of distributions of

root-mean-square fluctuations (RMSFs) of individual

basepairs within the bundle (Figure 4c). Obtaining such

information experimentally is currently impossible. While

agreement between model predictions and experiment are

apparent, flexibility predictions are likely to represent a

lower bound on true flexibility because they neglect

unknown structural defects in DNA origami structures

such as missing staple strands. Notwithstanding, the

ability to predict flexibility of DNA-based nanostructures

will be important to their design when specific local or

global flexibilities are required, including in the design of

monomers used to synthesize higher order polymers and

tile-like structures (22,23).

Molecular self-assembly using DNA and other nucleic

acids has great potential for building nanostructures for

target applications in materials and biological science. A

key challenge for leveraging this design modality in functional

applications is the ability to design target shapes to

meet desired geometrical and mechanical properties. The

present computational approach that enables shape and

mechanics prediction of DNA-based nanostructures

assembled using principles of DNA origami demonstrates

the capacity to predict not only solution shape but

also structural flexibility, or mechanical integrity, at the

nanoscale. We present here a new model for DNA origami

that includes the ability to model wireframe structures,

backbone nicks and single-stranded DNA used to design

tensegrity structures. In addition to predicting complex

3D solution shape and flexibility that is validated

experimentally for the first time here, our model reveals

novel, subtle structural features of DNA origami objects

including global out-of-plane deformations that require

further experimental validation. These new modeling

features are included in our online, web-based analysis

tool CanDo (http://cando.dna-origami.org). We anticipate

this predictive computational modeling framework

will make important contributions to the future development

of functional DNA nanostructures with applications

in diverse areas of science and technology.

SUPPLEMENTARY DATA

Supplementary Data are available at NAR Online:

Supplementary Note 1, Supplementary Figures S1–S23,

and Supplementary References [24–25].

ACKNOWLEDGEMENTS

Discussions with Klaus-Ju¨ rgen Bathe (M.I.T.) and Arash

Mahdavi (ADINA R&D, Inc.) are gratefully

acknowledged. We thank Yonggang Ke and Tim Liedl

for providing the caDNAno design files of 3D blocks on

a square lattice and a tensegrity structure, respectively,

and Shawn Douglas for help in interpreting the

caDNAno design file format.

FUNDING

MIT Faculty Start-up Funds and the Samuel A. Goldblith

Professorship (to M.B.); the Cluster for Integrated Protein

Science Munich and a Hans-Fischer tenure track grant

from TUM Institute for Advanced Study (to H.D.); a

stipend from the TUM graduate school ‘Materials

at Complex Interfaces’(CompInt to F.K.); The Cluster

for Integrated Protein Science Munich and Technische

Universita¨ t Mu¨ nchen Institute for Advanced Study

by the German Excellence Initiative; CompInt is funded

by the state of Bavaria. Funding for open access charge:

MIT Faculty Start-up Funds.

Conflict of interest statement. None declared.

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